SOLUTION: The bearing from A to C is S 52°E. the bearing from A to B is N 84°E. the bearing from B to C is S 38° W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the d
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-> SOLUTION: The bearing from A to C is S 52°E. the bearing from A to B is N 84°E. the bearing from B to C is S 38° W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the d
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Question 1200155: The bearing from A to C is S 52°E. the bearing from A to B is N 84°E. the bearing from B to C is S 38° W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the distance from A to C Answer by math_tutor2020(3817) (Show Source):
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A compass bearing of something like S 52°E means we start facing directly south, then we turn 52 degrees toward the east.
This is one way to draw out the diagram.
The diagram is to scale. It was made with GeoGebra.
The distance AB = 600 is due to the scratch work calculation shown below
distance = rate*time
distance = (250 mph)*(2.4 hrs)
distance = 600 miles
Angles EAC, DAB, and GBC are the compass bearings given to us. They are marked in blue shown above.
Angle
Compass Bearing
EAC = 52
S 52°E
DAB = 84
N 84°E
GBC = 38
S 38°W
The last two letters of each angle name tells us where we are located and where we are looking at in that exact order.
Example: Angle EAC has "A" and "C" as the last two letters. We're located at A and look toward C.
Vertical line DE is parallel to vertical line FG.
Segment AB is a transversal cut to these parallel lines.
Same side interior angles DAB and FBA are supplementary (because of the parallel lines).
So,
(angle DAB)+(angle FBA) = 180
(84)+(angle FBA) = 180
angle FBA = 180-84
angle FBA = 96
Angle BAC is found by noticing angles EAC, BAC, and DAB are supplementary
In other words,
(angle EAC)+(angle BAC)+(angle DAB) = 180
and similarly
(angle GBC)+(angle CBA)+(angle FBA) = 180
these two equations help determine angles BAC and CBA (44 and 46 respectively)
Then we use the idea that the inner angles of a triangle add to 180
A+B+C = 180
44+46+C = 180
90+C = 180
C = 90
Triangle ABC is a right triangle.
Since ABC is a right triangle, we have two options using the trig ratios sine and cosine
Option 1
Option 2
sin(angle) = opposite/hypotenuse sin(angle ABC) = AC/AB sin(46) = x/600 x = 600*sin(46) x = 431.60388020319 x = 431.6
cos(angle) = adjacent/hypotenuse cos(angle BAC) = AC/AB cos(44) = x/600 x = 600*cos(44) x = 431.60388020319 x = 431.6
Make sure your calculator is in degree mode.
Here's one possible route if you wanted to use the law of sines
sin(B)/b = sin(C)/c
sin(46)/x = sin(90)/600
600*sin(46) = x*sin(90)
x = 600*sin(46)/sin(90)
x = 431.60388020319
x = 431.6
Keep in mind that sin(90) = 1, so this calculation isn't entirely new compared to option 1 mentioned earlier.
Answer: Approximately 431.6 miles
Round this value however your teacher instructs.
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